Exercise 18, § 2

Let X be a locally compact space endowed with a bounded positive measure $\mu$. For every Banach space F, let $S(X, \mu; F)$ denote the space of classes of $\mu$-measurable mappings from X into F modulo the relation of equality $\mu$-almost everywhere, endowed with the topology of convergence in measure (INT, IV, p. 194, § 5, n$^o11$). Let $\mathscr{A}$ denote the commutative subalgebra of $\mathscr{L}(L^2_{\mathbf{C}}(X, \mu))$ formed by the multiplication endomorphisms by functions $f\in \mathscr{L}_{\mathbf{C}}^{\infty}(X, \mu)$.

Let $e:\mathbf{R}\rightarrow \mathbf{C}$ denote the homomorphism $t\mapsto$ exp(2$i\pi t$).

Let E be a topological vector space.

a) Let $f\in S(X, \mu;\mathbf{R})$ be such that $|f(x)|\geqslant 1$ for $\mu$-almost all $x\in X$. There exists a real number $t\in ]0,1[$ such that

b) Let $u$ be a linear mapping from E into $S(X, \mu;\mathbf{R})$. For every $x\in E$, let $\varrho_u(x)$ denote the element of $\mathscr{A}$ given by the multiplication endomorphism by the function $e\circ u(x)$, which belongs to $\mathscr{L}_{\mathbf{C}}^{\infty}(X, \mu)$. The mapping $\varrho_u$ is a continuous representation of G in $L^2_{\mathbf{C}}(X, \mu)$ if and only if the mapping $u$ is continuous. (If $u$ is not continuous, use the preceding question to verify that $\varrho$ is not continuous.)

c) Let $\varrho$ be a unitary representation of E in $L^2(X, \mu)$ whose image is contained in $\mathscr{A}$. There exists a unique continuous linear mapping $u: E\rightarrow$ $S(X, \mu;\mathbf{R})$ such that $\varrho =\varrho_u$. (First consider the case where $E =\mathbf{R}$, and then apply Stone's theorem.)